Manifolds with bounded integral curvature and no positive eigenvalue lower bounds

TL;DR

The paper constructs a sequence of closed surfaces with bounded diameter and curvature norms that lack a positive lower bound on the Laplacian's first eigenvalue, highlighting the necessity of small curvature norms for certain spectral estimates.

Contribution

It provides explicit examples demonstrating the necessity of small curvature norms for Lichnerowicz and Zhong-Yang type eigenvalue bounds under integral curvature conditions.

Findings

Sequences with bounded diameter and curvature norms can have arbitrarily small first eigenvalues

Smallness of the $L^p$ curvature norm is essential for spectral lower bounds

Counterexamples show limits of existing eigenvalue estimates under integral curvature bounds

Abstract

We provide an explicit construction of a sequence of closed surfaces with uniform bounds on the diameter and on $L^p$ norms of the curvature, but without a positive lower bound on the first non-zero eigenvalue of the Laplacian $\lambda_1$. This example shows that the assumption of smallness of the $L^p$ norm of the curvature is a necessary condition to derive Lichnerowicz and Zhong-Yang type estimates under integral curvature conditions.